Theorem axnul

Description: The Null Set Axiom of ZF set theory: there exists a set with no elements. Axiom of Empty Set of Enderton p. 18. In some textbooks, this is presented as a separate axiom; here we show it can be derived from Separation ax-sep . This version of the Null Set Axiom tells us that at least one empty set exists, but does not tell us that it is unique - we need the Axiom of Extensionality to do that (see nulmo ).

This proof, suggested by Jeff Hoffman, uses only ax-4 and ax-gen from predicate calculus, which are valid in "free logic" i.e. logic holding in an empty domain (see Axiom A5 and Rule R2 of LeBlanc p. 277). Thus, our ax-sep implies the existence of at least one set. Note that Kunen's version of ax-sep (Axiom 3 of Kunen p. 11) does not imply the existence of a set because his is universally closed, i.e., prefixed with universal quantifiers to eliminate all free variables. His existence is provided by a separate axiom stating E. x x = x (Axiom 0 of Kunen p. 10).

See axnulALT for a proof directly from ax-rep .

This theorem should not be referenced by any proof. Instead, use ax-nul below so that the uses of the Null Set Axiom can be more easily identified. (Contributed by Jeff Hoffman, 3-Feb-2008) (Revised by NM, 4-Feb-2008) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	axnul	$$⊢\exists x\forall y\neg y\in x$$

Proof

Step	Hyp	Ref	Expression
1		ax-sep	$$⊢\exists x\forall y\left(y\in x\leftrightarrow \left(y\in z\wedge \perp \right)\right)$$
2		fal	$$⊢\neg \perp$$
3	2	intnan	$$⊢\neg \left(y\in z\wedge \perp \right)$$
4		id	$$⊢\left(y\in x\leftrightarrow \left(y\in z\wedge \perp \right)\right)\to \left(y\in x\leftrightarrow \left(y\in z\wedge \perp \right)\right)$$
5	3 4	mtbiri	$$⊢\left(y\in x\leftrightarrow \left(y\in z\wedge \perp \right)\right)\to \neg y\in x$$
6	5	alimi	$$⊢\forall y\left(y\in x\leftrightarrow \left(y\in z\wedge \perp \right)\right)\to \forall y\neg y\in x$$
7	1 6	eximii	$$⊢\exists x\forall y\neg y\in x$$